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Describing Quasi-Graphic Matroids

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Describing Quasi-Graphic Matroids DESCRIBING QUASI-GRAPHIC MATROIDS NATHAN BOWLER∗, DARYL FUNKy, AND DANIEL SLILATYz Abstract. The class of quasi-graphic matroids recently introduced by Geelen, Gerards, and Whittle generalises each of the classes of frame matroids and lifted- graphic matroids introduced earlier by Zaslavsky. For each biased graph (G; B) Zaslavsky defined a unique lift matroid L(G; B) and a unique frame matroid F (G; B), each on ground set E(G). We show that in general there may be many quasi-graphic matroids on E(G) and describe them all: for each graph G and partition (B; L; F) of its cycles such that B satisfies the theta property and each cycle in L meets each cycle in F, there is a quasi-graphic matroid M(G; B; L; F) on E(G). Moreover, every quasi-graphic matroid arises in this way. We provide cryptomorphic descriptions in terms of subgraphs corresponding to circuits, co- circuits, independent sets, and bases. Equipped with these descriptions, we prove some results about quasi-graphic matroids. In particular, we provide alternate proofs that do not require 3-connectivity of two results of Geelen, Gerards, and Whittle for 3-connected matroids from their introductory paper: namely, that every quasi-graphic matroid linearly representable over a field is either lifted- graphic or frame, and that if a matroid M has a framework with a loop that is not a loop of M then M is either lifted-graphic or frame. We also provide sufficient conditions for a quasi-graphic matroid to have a unique framework. Zaslavsky has asked for those matroids whose independent sets are contained in the collection of independent sets of F (G; B) while containing those of L(G; B), for some biased graph (G; B). Adding a natural (and necessary) non-degeneracy condition defines a class of matroids, which we call biased-graphic. We show that the class of biased-graphic matroids almost coincides with the class of quasi- graphic matroids: every quasi-graphic matroid is biased-graphic, and if M is a biased-graphic matroid that is not quasi-graphic then M is a 2-sum of a frame matroid with one or more lifted-graphic matroids. 1. Context and motivation In series of foundational papers [8, 9, 10, 11, 12] Thomas Zaslavsky introduced arXiv:1808.00489v2 [math.CO] 24 Oct 2019 biased graphs, their associated frame and lift matroids, and established their basic properties. A matroid is a frame matroid if it may be extended so that it has a basis B such that every element is spanned by at most two elements of B. Such a basis is a frame for the matroid. A matroid M is a lift (or lifted-graphic) matroid if it is an elementary lift of a graphic matroid; that is, if there is a matroid N with E(N) = E(M) [ feg such that Nne = M and N=e is graphic. These are fundamental and important classes of matroids. Frame matroids were introduced by Zaslavsky as a significant generalisation of Dowling geometries (the Date: October 24, 2019. 1 2 BOWLER, FUNK, AND SLILATY cycle matroid of a complete graph is a Dowling geometry over the trivial group). Moreover, classes of representable frame matroids play an important role in the matroid minors project of Geelen, Gerards, and Whittle [3, Theorem 3.1], analogous to that of graphs embedded on surfaces in graph structure theory. Geelen, Gerards, and Whittle recently introduced the class of quasi-graphic ma- troids as a common generalisation of each of these classes [4]. A matroid M is quasi-graphic if it has a framework: that is, a graph G with (i) E(G) = E(M), such that (ii) the rank of the edge set of each component of G is at most the size of its vertex set, (iii) for each vertex v 2 V (G) the closure in M of E(G − v) does not contains an edge with endpoints v; w with w 6= v, and (iv) no circuit of M induces a subgraph in G of more than two components. For each quasi-graphic matroid M there is a biased graph (G; B), where G is a framework for M and B is the set of cycles of G that are circuits of M. However, there is not enough information provided by a biased graph to determine a quasi-graphic matroid. We define two notions that provide the missing information: bracelet functions and proper tripari- tions.A bracelet is a vertex disjoint pair of unbalanced cycles in a biased graph. A bracelet function χ is a function that maps each bracelet of a biased graph to fdependent; independentg. If χ obeys a certain condition then we say χ is proper and we may define a matroid M(G; B; χ) on E(G), in which the independence of each bracelet is given by χ. Given a graph G, a proper tripartition of the cycles of G is a partition (B; L; F) of its collection of cycles such that B obeys the theta property and every cycle in L meets every cycle in F. Thus in a proper tripartition every bracelet either has both its cycles in L or both its cycles in F. Given a graph G together with a proper tripartition (B; L; F) of its cycles, we may define a matroid M(G; B; L; F) on E(G), in which a bracelet is independent precisely when both of its cycles are in F. We can now state our main result. Theorem 1.1. Let M be a matroid and let (G; B) be a biased graph with E(G) = E(M). The following are equivalent. (1) There is a proper bracelet function χ for G such that M = M(G; B; χ). (2) There is a proper tripartition (B; L; F) of the cycles of G such that M = M(G; B; L; F). (3) M is quasi-graphic with framework G and B is the set of cycles of G that are circuits of M. 1.1. Frame matroids. In [10] Zaslavsky showed that the class of frame matroids is precisely that of matroids arising from biased graphs, as follows. Let M be a frame matroid on ground set E, with frame B. By adding elements in parallel if necessary, we may assume B \ E = ;. Hence for some matroid N, M = NnB where B is a basis for N and every element e 2 E is minimally spanned by either a single element or a pair of elements in B. Let G be the graph with vertex set B and edge set E, in which e is a loop with endpoint f if e is parallel with f 2 B, and otherwise e is an edge with endpoints f; f 0 2 B if e 2 clff; f 0g. The edge set of a cycle of G is either independent or a circuit in M. A cycle C in G whose edge set is a circuit of M is said to be balanced; otherwise C is unbalanced. Thus the cycles of G are DESCRIBING QUASI-GRAPHIC MATROIDS 3 partitioned into two sets: those that are circuits and those that are independent. The collection of balanced cycles of G is denoted B. The bias of a cycle is given by the set of the bipartition to which it belongs. Together the pair (G; B) is a biased graph. We say such a biased graph (G; B) represents the frame matroid M, and we write M = F (G; B). The circuits of a frame matroid M may be precisely described in terms of biased subgraphs of such a biased graph (G; B). A theta graph consists of a pair of distinct vertices with three internally disjoint paths between them. The circuits of M are precisely those sets of edges inducing one of: a balanced cycle, a theta subgraph in which all three cycles are unbalanced, two edge-disjoint unbalanced cycles inter- secting in exactly one vertex, or two vertex-disjoint unbalanced cycles along with a minimal path connecting them. The later two biased subgraphs are called hand- cuffs, tight or loose, respectively. It is a straightforward consequence of the circuit elimination axiom that a biased theta subgraph of (G; B) may not contain exactly two balanced cycles. We call this the theta property. Zaslavsky further showed [10] that conversely, given any graph G and partition (B; U) of its cycles, all that is required for there to exist a frame matroid whose circuits are given by the collection of biased subgraphs described above is that the collection B satisfy the theta property. 1.2. Lifted-graphic matroids. Let N be a matroid on ground set E [ feg, and suppose G is a graph with edge set E and with cycle matroid M(G) equal to N=e. Then M = Nne is a lifted-graphic matroid. Each cycle in G is either a circuit of N, and so of M, or together with e forms a circuit of N. Thus again the cycles of G are naturally partitioned into two sets: those that are circuits of M and those that are independent in M; thus a lifted-graphic matroid naturally gives rise to a biased graph. Again, a cycle whose edge set is a circuit of M is said to be balanced, and those whose edges form an independent set are unbalanced. In [9] Zaslavsky showed that the circuits of M are precisely those sets of edges in G inducing one of: a balanced cycle, a theta subgraph in which all three cycles are unbalanced, two edge disjoint unbalanced cycles meeting in exactly one vertex, or a pair of vertex-disjoint unbalanced cycles.
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